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## CSE 143 Lecture 17

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### CSE 143Lecture 17

Recursive Backtracking

slides created by Marty Stepp

http://www.cs.washington.edu/143/

ideas and examples taken from Stanford University CS slides/lectures

Exercise: Permutations

- Write a method permute that accepts a string as a parameter and outputs all possible rearrangements of the letters in that string. The arrangements may be output in any order.
- Example:permute("MARTY")outputs the followingsequence of lines:

Examining the problem

- Think of each permutation as a set of choices or decisions:
- Which character do I want to place first?
- Which character do I want to place second?
- ...
- solution space: set of all possible sets of decisions to explore
- We want to generate all possible sequences of decisions.

for (each possible first letter):

for (each possible second letter):

for (each possible third letter):

...

print!

- This is called a depth-first search

Backtracking

- backtracking: A general algorithm for finding solution(s) to a computational problem by trying partial solutions and then abandoning them ("backtracking") if they are not suitable.
- a "brute force" algorithmic technique (tries all paths; not clever)
- often (but not always) implemented recursively

Applications:

- producing all permutations of a set of values
- parsing languages
- games: anagrams, crosswords, word jumbles, 8 queens
- combinatorics and logic programming

Backtracking algorithms

A general pseudo-code algorithm for backtracking problems:

explore(choices):

- if there are no more choices to make: stop.
- else:
- Make a single choice C from the set of choices.
- Remove C from the set of choices.
- explore the remaining choices.
- Un-make choice C.
- Backtrack!

Backtracking strategies

- When solving a backtracking problem, ask these questions:
- What are the "choices" in this problem?
- What is the "base case"? (How do I know when I'm out of choices?)
- How do I "make" a choice?
- Do I need to create additional variables to remember my choices?
- Do I need to modify the values of existing variables?
- How do I explore the rest of the choices?
- Do I need to remove the made choice from the list of choices?
- Once I'm done exploring the rest, what should I do?
- How do I "un-make" a choice?

Permutations revisited

- Write a method permute that accepts a string as a parameter and outputs all possible rearrangements of the letters in that string. The arrangements may be output in any order.
- Example:permute("MARTY")outputs the followingsequence of lines:

Exercise solution

// Outputs all permutations of the given string.

public static void permute(String s) {

permute(s, "");

}

private static void permute(String s, String chosen) {

if (s.length() == 0) {

// base case: no choices left to be made

System.out.println(chosen);

} else {

// recursive case: choose each possible next letter

for (int i = 0; i < s.length(); i++) {

char c = s.charAt(i); // choose

s = s.substring(0, i) + s.substring(i + 1);

chosen += c;

permute(s, chosen);// explore

s = s.substring(0, i) + c + s.substring(i + 1);

chosen = chosen.substring(0, chosen.length() - 1);

} // un-choose

}

}

Exercise solution 2

// Outputs all permutations of the given string.

public static void permute(String s) {

permute(s, "");

}

private static void permute(String s, String chosen) {

if (s.length() == 0) {

// base case: no choices left to be made

System.out.println(chosen);

} else {

// recursive case: choose each possible next letter

for (int i = 0; i < s.length(); i++) {

String ch = s.substring(i, i + 1); // choose

String rest = s.substring(0, i) + // remove

s.substring(i + 1);

permute(rest, chosen + ch);// explore

}

} // (don't need to "un-choose" because

} // we used temp variables)

Exercise: Combinations

- Write a method combinations that accepts a string s and an integer k as parameters and outputs all possible k -letter words that can be formed from unique letters in that string. The arrangements may be output in any order.
- Example:combinations("GOOGLE", 3)outputs the sequence oflines at right.
- To simplify the problem, you may assumethat the string s contains at least kunique characters.

Initial attempt

public static void combinations(String s, int length) {

combinations(s, "", length);

}

private static void combinations(String s, String chosen, int length) {

if (length == 0) {

System.out.println(chosen); // base case: no choices left

} else {

for (int i = 0; i < s.length(); i++) {

String ch = s.substring(i, i + 1);

if (!chosen.contains(ch)) {

String rest = s.substring(0, i) + s.substring(i + 1);

combinations(rest, chosen + ch, length - 1);

}

}

}

}

- Problem: Prints same string multiple times.

Exercise solution

public static void combinations(String s, int length) {

Set<String> all = new TreeSet<String>();

combinations(s, "", all, length);

for (String comb : all) {

System.out.println(comb);

}

}

private static void combinations(String s, String chosen,

Set<String> all, int length) {

if (length == 0) {

all.add(chosen); // base case: no choices left

} else {

for (int i = 0; i < s.length(); i++) {

String ch = s.substring(i, i + 1);

if (!chosen.contains(ch)) {

String rest = s.substring(0, i) + s.substring(i + 1);

combinations(rest, chosen + ch, all, length - 1);

}

}

}

}

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